Tangent planes for F(x, y, z)=0 Find an equation of the...

Tangent planes for $F(x, y, z)=0$ Find an equation of the plane tangent to the following surfaces at the given points. $$z^{2}-x^{2} / 16-y^{2} / 9-1=0 ;(4,3,-\sqrt{3}) \text { and }(-8,9, \sqrt{14})$$

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Key Concepts

Implicit Surface

An implicit surface is defined by an equation of the form F(x, y, z) = 0 rather than by solving explicitly for one variable. This representation is useful in many applications because it allows complex surfaces to be described without needing to isolate a single variable and can include surfaces that may not be functions in the traditional sense.

Gradient Vector

The gradient vector of a function F(x, y, z) is the vector of its partial derivatives and is given by ?F = (F_x, F_y, F_z). It points in the direction of the greatest rate of increase of the function. An important property of the gradient is that it is perpendicular (normal) to the level surface defined by F(x, y, z) = 0 at any point.

Tangent Plane

The tangent plane to an implicit surface at a given point can be determined using the gradient vector. Since the gradient is normal to the surface at that point, the equation of the tangent plane can be written as ?F(x?, y?, z?) · (x - x?, y - y?, z - z?) = 0, where (x?, y?, z?) is the point of tangency. This formula is fundamental for approximating the surface locally around the point.

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